superalgebra and (synthetic ) supergeometry
Supergeometry is the (higher) geometry over the base topos on superpoints modeled on the canonical line object $\mathbb{R}$ in there.
As ordinary differential geometry studies spaces – smooth manifolds – that locally look like vector spaces, supergeometry studies spaces – supermanifolds – that locally look like super vector spaces.
As ordinary algebraic geometry studies spaces – schemes – that locally look like affine spaces, supergeometry studies superschemes.
From the point of view of noncommutative geometry, the supergeometry is a very mild special case of noncommutativity in geometry: some coordinates commute, some anticommute.
For more see at geometry of physics -- supergeometry.
$\phantom{A}$(higher) geometry$\phantom{A}$ | $\phantom{A}$site$\phantom{A}$ | $\phantom{A}$sheaf topos$\phantom{A}$ | $\phantom{A}$∞-sheaf ∞-topos$\phantom{A}$ |
---|---|---|---|
$\phantom{A}$discrete geometry$\phantom{A}$ | $\phantom{A}$Point$\phantom{A}$ | $\phantom{A}$Set$\phantom{A}$ | $\phantom{A}$Discrete∞Grpd$\phantom{A}$ |
$\phantom{A}$differential geometry$\phantom{A}$ | $\phantom{A}$CartSp$\phantom{A}$ | $\phantom{A}$SmoothSet$\phantom{A}$ | $\phantom{A}$Smooth∞Grpd$\phantom{A}$ |
$\phantom{A}$formal geometry$\phantom{A}$ | $\phantom{A}$FormalCartSp$\phantom{A}$ | $\phantom{A}$FormalSmoothSet$\phantom{A}$ | $\phantom{A}$FormalSmooth∞Grpd$\phantom{A}$ |
$\phantom{A}$supergeometry$\phantom{A}$ | $\phantom{A}$SuperFormalCartSp$\phantom{A}$ | $\phantom{A}$SuperFormalSmoothSet$\phantom{A}$ | $\phantom{A}$SuperFormalSmooth∞Grpd$\phantom{A}$ |
Some historically influential general considerations are in
Introductory lecture notes include
Yuri Manin, chapter 4 of Gauge Field Theory and Complex Geometry, Grundlehren der Mathematischen Wissenschaften 289, Springer 1988
L. Caston, R. Fioresi, Mathematical Foundations of Supersymmetry (arXiv:0710.5742)
Gennadi Sardanashvily, Lectures on supergeometry (arXiv:0910.0092)
The observation that the study of super-structures in mathematics is usefully regarded as taking place over the base topos on the site of super points has been made around 1984 in
Albert Schwarz, On the definition of superspace, Teoret. Mat. Fiz. (1984) Volume 60, Number 1, Pages 37–42, (russian original pdf)
Alexander Voronov, Maps of supermanifolds , Teoret. Mat. Fiz. (1984) Volume 60, Number 1, Pages 43–48
and in
A summary/review is in the appendix of
Anatoly Konechny and Albert Schwarz,
On $(k \oplus l|q)$-dimensional supermanifolds in Supersymmetry and Quantum Field Theory (D. Volkov memorial volume) Springer-Verlag, 1998 , Lecture Notes in Physics, 509 , J. Wess and V. Akulov (editors)(arXiv:hep-th/9706003)
Theory of $(k \oplus l|q)$-dimensional supermanifolds Sel. math., New ser. 6 (2000) 471 - 486
Albert Schwarz, I- Shapiro, Supergeometry and Arithmetic Geometry (arXiv:hep-th/0605119)
A review of all this as geometry in the topos over the category of superpoints is in
Formulation in terms of synthetic differential supergeometry is in
For many more references see at supermanifold.
Plenty of discussion of supergeometry with an eye towards supersymmetry in quantum field theory is in
especially in the contribution
The appendix there
means to sort out various sign conventions of relevance.
Discussion of how supersymmetry is universally induced in higher category theory/homotopy theory by the free abelian ∞-group on the point – the sphere spectrum – is in
For more on this see at superalgebra.
Discussion related to G-structure and Killing spinors includes
Discussion of classical field theory with fermions as taking place on supermanifolds is in the following references
Pierre Deligne, Daniel Freed, Classical field theory (1999) (pdf)
this is a chapter in
P. Deligne, P. Etingof, D.S. Freed, L. Jeffrey, D. Kazhdan, J. Morgan, D.R. Morrison, E. Witten (eds.) Quantum Fields and Strings, A course for mathematicians, 2 vols. Amer. Math. Soc. Providence 1999. (web version)
Daniel Freed, Classical field theory and Supersymmetry, IAS/Park City Mathematics Series Volume 11 (2001) (pdf)
Giovanni Giachetta, Luigi Mangiarotti, Gennadi Sardanashvily, chapter 3 of Advanced classical field theory, World Scientific (2009)
Gennadi Sardanashvily, Grassmann-graded Lagrangian theory of even and odd variables (arXiv:1206.2508)
Gennadi Sardanashvily, Noether’s Theorems: Applications in Mechanics and Field Theory, Studies in Variational Geometry, 2016
Last revised on June 25, 2018 at 09:06:47. See the history of this page for a list of all contributions to it.